P01 — Orbit Processing Unit (OPU)
| Field | Value |
|---|---|
| Domain | Computing Architectures |
| System | RPU with tape = Gr(k,n) |
| Group | U(n) acting on Gr(k,n) |
| H^k tier | H⁰ / H¹ / H² |
| ISA | Forge (β ≈ β*) |
| Status | Validated |
| Opcodes | ORBIT · TWIST · BIND · SPLIT · SPLAT · LABEL |
| Papers | Paper 598, Paper 205, Paper 489 |
Physical system
The Orbit Processing Unit (OPU) is a Resonance Processing Unit (RPU, Paper 205) whose computational tape is the Grassmannian Gr(k,n) = U(n)/(U(k)×U(n−k)) — the space of k-dimensional subspaces of ℂⁿ. The head state is a k-plane V ∈ Gr(k,n); the alphabet is the Schubert cell decomposition of Gr(k,n) (C(n,k) cells); the memory register is the NOON spectrum {σ₁²,…,σₖ²} (singular values of the head’s coordinate matrix); the halting condition is Schubert variety crossing — the β* snap event where σ₁² crosses a critical threshold.
The OPU is not an abstraction: FeMoco, the ribosome, and photosynthetic FMO are all physical OPUs, running the Forge ISA on biological Grassmannians at 300 K.
Target category
Symp(Gr) — the category of Grassmannian-fibred symplectic manifolds, with the U(n) action, the Fubini-Study metric, and the Schubert stratification. Morphisms are U(n)-equivariant symplectic maps (geodesics on Gr(k,n)).
Interpretation functor
F: C → Symp(Gr) defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Geodesic step on Gr(k,n): V → exp(tX)·V for X ∈ u(n); winding number = Schubert cell index traversed |
| TWIST | Berry phase accumulation: φ = i∮⟨V|dV⟩ around a closed loop in Gr(k,n); H¹ holonomy |
| BIND | H² holonomy: non-abelian parallel transport around a 2-cycle in Gr(k,n); requires quantum hardware |
| SPLIT | Schmidt decomposition: V → U Σ W† (SVD); outputs NOON spectrum {σₖ²} |
| SPLAT | Reverse Schmidt: reconstruct V from NOON spectrum (CI coefficient update) |
| LABEL | Schubert cell index: read current cell from σ₁²; halt flag if σ₁² > 0.88 |
ISA programme
INIT: LABEL[V₀ | HF reference, σ₁²=1] -- start at Hartree-Fock Schubert cell
SPLIT: SPLIT[V₀ → U₀ Σ₀ W₀†] -- read initial NOON spectrum
STEP: ORBIT[V → exp(tX)·V | X = orbital gradient] -- geodesic step (CASSCF iteration)
PHASE: TWIST[φ += i⟨V|dV⟩] -- accumulate Berry phase
HALT?: LABEL[σ₁² < 0.88?] -- Schubert variety crossing = β* snap
BIND: BIND[holonomy | H² pairs only] -- non-abelian correction (quantum hardware)
READ: SPLAT[{σₖ²}, {φₖ}] -- output NOON spectrum + Berry phases
Computable output
- NOON spectrum {σₖ²}: the C/T skeleton classification (C-boxes: σₖ² ≈ 0 or 2; T-arrows: σₖ² ∈ (0,2)). Directly interpretable as the alchemi output.
- Halt condition: σ₁² crosses the Schubert variety boundary at σ₁² ≈ 0.88 — the β* snap. Corresponds to the CASSCF convergence criterion in chemical calculations (verified in Paper 596, x596c).
- Berry phases {φₖ}: H¹ holonomy per T-arrow pair. Equal to the CASSCF orbital rotation angles at convergence.
- Correlated energy: E = ⟨Ψ|H|Ψ⟩ at the halted k-plane V. The geodesic length d_FS(V₀, V) is the geometric measure of correlation difficulty.
The RPU family
| XPU | Tape manifold | Group | β regime |
|---|---|---|---|
| QPU | CP^{2ⁿ⁻¹} (full Hilbert space) | U(2ⁿ) | β = it |
| OPU | Gr(k,n) | U(n) | finite β |
| PPU | Bruhat-Tits tree | PGL(2,ℚₚ) | p-adic β |
| Fano OPU | Gr(3,7), G₂ symmetry | G₂ | finite β |
GPU boundary
| Tier | Opcode | GPU emulable? | Notes |
|---|---|---|---|
| H⁰ | ORBIT + SPLIT/SPLAT | Yes | batched SVD + GEMM (cuBLAS) |
| H¹ | TWIST | Yes | complex phase det(U†V) |
| H² | BIND | No | requires phase-coherent quantum hardware |
Existing CASSCF codes (PySCF, GAMESS, ORCA with GPU) are implicit OPU emulators — they compute geodesics on Gr(k,n) without naming them as such.
Natural OPU instances
| System | Gr(k,n) | Halt condition | Paper |
|---|---|---|---|
| FeMoco (nitrogenase) | Gr(3,7) | N₂ → 2NH₃ committed | 488 |
| DNA Polymerase III | Gr(H⁰×H¹×H²) | Base-pair confirmed | 510 |
| FMO complex | Gr(1,7) | Exciton at RC | 325 |
| Ribosome | Gr(3,6) | Peptide bond formed | 510 |
Validation
- CASSCF orbital gradient = covariant derivative on Gr(k,n): confirmed to 6 decimal places (Paper 594).
- Schubert variety crossing = β* snap: confirmed numerically for N₂ and Fe²⁺ (Paper 596, x596c, SHA d4a5d465).
- FeMoco as Fano OPU at 300 K: Papers 488/310, validated against EPR spectrum.